The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
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(DS)-WEAK COMMUTATIVITY CONDITION AND COMMON FIXED POINT IN INTUITIONISTIC MENGER SPACES

  • Received : 2010.07.03
  • Accepted : 2011.06.02
  • Published : 2011.08.31
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Abstract

The aim of this paper is to define a new commutativity condition for a pair of self mappings i.e., (DS)-weak commutativity condition, which is weaker that compatibility of mappings in the settings of intuitionistic Menger spaces. We show that a common fixed point theorem can be proved for nonlinear contractive condition in intuitionistic Menger spaces without assuming continuity of any mapping. To prove the result we use (DS)-weak commutativity condition for mappings. We also give examples to validate our results.

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References

  1. Amini M. & Saadati R.: Some properties of continuous t-norms and s-norms. Int. J Pure Appl Math. 16 (2004), 157-164.
  2. Boyd D.W. & Wong J.S.W.: On nonlinear contractions. Proc. Amer. Math. Soc. 20 (1969), 458-464. https://doi.org/10.1090/S0002-9939-1969-0239559-9
  3. Bylka C.: Fixed point theorems of Matkowski on probabilistic metric spaces. Demonstratio Math. 29 (1996), 158-164.
  4. Cho Y.J., Murthy P.P. & Stojakovic M.: Compatible mappings of type (A) and common fixed points in Menger spaces. Com. Korean Math. J. 7 (1992), 325-339.
  5. Civic L.B.: Milovanovic, Arandjelovic M.M.: Common fixed point theorem for R-weak commuting mappings in Menger spaces. J. Indian Acad. Math. 22 (2000), 199-210.
  6. Dedeic R. & Sarapa N.: A common fixed point theorem for three mappings on Menger spaces. Math. Japanica 34 (1989), no. 6, 919-923.
  7. Deshpande B.: Fixed point and (DS)-weak commutativity condition in intuitionistic fuzzy metric space. Chaos, Solitons & Fractals 42 (2009), 2722-2728. https://doi.org/10.1016/j.chaos.2009.03.178
  8. EI Naschie M.S.: On the uncertainly of cantorian geometry and the two-slit experiment. Chaos, Solitons and Fractals 9 (1998), no. 3, 517-529. https://doi.org/10.1016/S0960-0779(97)00150-1
  9. EI Naschie M.S.: On the verification of heterotic strings theory and $e^{(\infty)}$ theory. Chaos, Solitons and Fractals 11 (2000), 2397-2408. https://doi.org/10.1016/S0960-0779(00)00108-9
  10. EI Naschie M.S.: A review of applications and results of E-infinity theory. Int J. Nonlinear Sci Numer Simulet 8 (2007), no. 1, 11-20.
  11. George A. & Veeramani P.: On some result in fuzzy metric space. Fuzzy Sets and Systems 64 (1994), 395-399. https://doi.org/10.1016/0165-0114(94)90162-7
  12. George A. & Veeramani P.: On some result of analysis for fuzzy metric spaces. Fuzzy Sets and Systems 90 (1997), 365-368. https://doi.org/10.1016/S0165-0114(96)00207-2
  13. Gregori V. & Romaguera S.: Characterzing completable fuzzy metric spaces. Fuzzy Sets and Systems 144 (2004), 411-420. https://doi.org/10.1016/S0165-0114(03)00161-1
  14. Gregori V. & Romaguera S.: On completion of fuzzy metric spaces. Fuzzy Sets and Systems 130 (2002), 399-404. https://doi.org/10.1016/S0165-0114(02)00115-X
  15. Gregori V. & Romaguera S.: Some properties of fuzzy metric spaces. Fuzzy Sets and Systems 115 (2000), 485-489. https://doi.org/10.1016/S0165-0114(98)00281-4
  16. Hadzic O.: A fixed point theorem in Menger spaces. Publ. Inst. Math. Beograd 20 (1979), 107-112.
  17. Hadzic O.: Some theorems on the fixed points in probabilistic metric and random normed spaces. Boll. Un. Mat. Ital. 13 (1981), no. 5, 1-11.
  18. Hadzic O.: Fixed point theory in probabilistic metric spaces. Novi Sad : Serbian Academy of Science and Arts, 1995.
  19. Hadzic O.: Fixed point theory in topological vector space. Novi Sad : University of Novi Sad, Institute of Mathematics, 1984.
  20. Hu C.: C-structure of FTS. V:Fuzzy metric spaces. J. Fuzzy Math. 3 (1995), 711-721.
  21. Jesic S.N. & Babacev N.A.: Common fixed point theorems in intuitionistic fuzzy metric space and L-fuzzy metric spaces with nonlinear contractive condition. Chaos Solitons and Fractals 37 (2008), 675-687. https://doi.org/10.1016/j.chaos.2006.09.048
  22. Jungck G. & Rhoades B.E.: Fixed point for set valued functions without continuity. Indian J. Pure Appl. Math. 29 (1998), no. 3, 227-238.
  23. Kubiaczyk I. & Deshpande B.: Comments on a fixed point theorem in Menger space through weak compatibility. Italian J. Pure and Appl. Math. 23 (2008), 137-146.
  24. Kutukcu S., Tuna A. & Yakut A.T.: Generalized contraction mapping principle in intuitionistic Menger spaces and application to differential equations. Applied Mathematics and Mechanics (English Edition) 28 (2007), no. 6, 799-809. https://doi.org/10.1007/s10483-007-0610-z
  25. Lowen R.: Fuzzy set theory. Dordrecht : Kluwer Academic Publishers, 1996.
  26. Menger K.: Statistical Metric. Proc Nat Acad Sci USA 28 (1942), 535-537. https://doi.org/10.1073/pnas.28.12.535
  27. Mishra S.N.: Common fixed points of compatible mappings in PM-spaces. Math. Japanica 36 (1991), no. 2, 283-289.
  28. O'Regan D. & Sadaati R.: Nonlinear contraction theorems in probabilistic spaces. Appl. Math. Comput. 195 (2008), 86-93. https://doi.org/10.1016/j.amc.2007.04.070
  29. Pant R.P.: Common fixed point of non commuting mappings. J. Math. Anal. Appl. 188 (1994), 436-40. https://doi.org/10.1006/jmaa.1994.1437
  30. Park J.H.: Intuitionistic fuzzy metric spaces. Chaos, Solitons & Fractals 22 (2004), 1039-1046. https://doi.org/10.1016/j.chaos.2004.02.051
  31. Pathak H.K. Cho Y.J. & Kang S.M.: Remarks on R-weakly commuting mappings and common fixed point theorems. Bull. Korean Math. Soc. 34 (1997), 247-257.
  32. Pathak H.K., Kang S.M. & Baek J.H.: Weak compatible mappings of type (A) and common fixed point in Menger spaces. Commun. Korean Math. Soc. 10 (1995), no. 1, 67-83.
  33. Radu V.: On some contraction principle in Menger spaces. An Univ. Timisoara, Stiinte Math. 22 (1984), 83-88.
  34. Radu V.: On some contraction type mappings in Menger spaces. An Univ. Timisoara, Stiinte Math. 23 (1985), 61-65.
  35. Rashwan R.A. & Hedar A.: On common fixed point theorems of compatible mappings in Menger spaces. Demonstratio Math. 31 (1998), no. 3, 537-546.
  36. Schweizer B. & Sklar A.: Probabilistic metric spaces. Elservier/North-Holland, New York, 1983.
  37. Sehgal V.M. & Bharucha-Reid A.T.: Fixed points of contraction mappings on probabilistic metric spaces. Math. System Theory 6 (1972), 97-102. https://doi.org/10.1007/BF01706080
  38. Serstnev A.N.: On the motion of a random normed space. Dokl, Akad, Nauk, SSSR 149 (1963), 280-283(English translation in Soviet Math, Dokl.) 4 (1963), 388-390.
  39. Sharma S. & Bagvan A.: Common fixed point theorem for six mappings in Menger space. Fasciculi Mathematici 37 (2007), 67-77.
  40. Sharma S. & Deshpande B.: Common fixed point theorems for weakly compatible mappings without continuity in Menger spaces. Pure Appl. Math. 10 (2003), no. 2, 133-144.
  41. Sharma S. & Deshpande B.: Common fixed point theorems for non-compatible mappings and Meir-Keeler type contractive condition in fuzzy metric space. International Review of Fuzzy Mathematics 1 (2006), no. 2, 147-159.
  42. Sharma S. & Deshpande B.: A fixed point theorem in intuitionistic fuzzy metric spaces by using new commutativity condition. Submitted for publication.
  43. Sharma S. & Deshpande B.: Compatible mappings of type (I) and (II) on intuitionistic fuzzy metric spaces in consideration of common fixed point. Commun. Korean Math. Soc. 24 (2009), no. 2, 197-214. https://doi.org/10.4134/CKMS.2009.24.2.197
  44. Sharma S., Deshpande B. & Tiwari R.: Common fixed point theorems for finite numbers of mappings without continuity and compatibility in Menger spaces. J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 15 (2008), no. 2, 135-151.
  45. Sherwood H.: Complete probabilistic metric spaces. Z. Wahrsch. Verw. Gebiete 20 (1971), 117-128. https://doi.org/10.1007/BF00536289
  46. Stojakovic M.: Common fixed point theorem in probabilistic metric spaces and its applications. Glasnik Mat. 23 (1988), 203-211.
  47. Stojakovic M.: Common fixed point theorems in complete metric and probabilistic metric spaces. Bull Austral. Math. Soc. 36 (1987), 73-88. https://doi.org/10.1017/S0004972700026319
  48. Stojakovic M.: Fixed point theorem in probabilistic metric spaces. Kobe J. Math. 2 (1985), 1-9.
  49. Vasuki R.: Common fixed point for R-weakly commuting maps in fuzzy metric space. Indian J. Pure Appl. Math. 30 (1999), no. 4, 419-423.